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Solve 3 2 x + 1 = 4 3 x + 1 . State the excluded values.

x = 7 17 . Excluded values are x = 1 2 and x = 1 3 .

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Solving a rational equation with factored denominators and stating excluded values

Solve the rational equation after factoring the denominators: 2 x + 1 1 x 1 = 2 x x 2 1 . State the excluded values.

We must factor the denominator x 2 −1. We recognize this as the difference of squares, and factor it as ( x 1 ) ( x + 1 ) . Thus, the LCD that contains each denominator is ( x 1 ) ( x + 1 ) . Multiply the whole equation by the LCD, cancel out the denominators, and solve the remaining equation.

( x 1 ) ( x + 1 ) [ 2 x + 1 1 x 1 ] = [ 2 x ( x 1 ) ( x + 1 ) ] ( x 1 ) ( x + 1 ) 2 ( x 1 ) 1 ( x + 1 ) = 2 x 2 x 2 x 1 = 2 x Distribute the negative sign . −3 x = 0 −3 = x

The solution is x = −3. The excluded values are x = 1 and x = −1.

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Solve the rational equation: 2 x 2 + 1 x + 1 = 1 x 2 x 2 .

x = 1 3

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Finding a linear equation

Perhaps the most familiar form of a linear equation is the slope-intercept form, written as y = m x + b , where m = slope and b = y −intercept . Let us begin with the slope.

The slope of a line

The slope    of a line refers to the ratio of the vertical change in y over the horizontal change in x between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.

m = y 2 y 1 x 2 x 1

If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in [link] . The lines indicate the following slopes: m = −3 , m = 2 , and m = 1 3 .

Coordinate plane with the x and y axes ranging from negative 10 to 10.  Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.

The slope of a line

The slope of a line, m , represents the change in y over the change in x. Given two points, ( x 1 , y 1 ) and ( x 2 , y 2 ) , the following formula determines the slope of a line containing these points:

m = y 2 y 1 x 2 x 1

Finding the slope of a line given two points

Find the slope of a line that passes through the points ( 2 , −1 ) and ( −5 , 3 ) .

We substitute the y- values and the x- values into the formula.

m = 3 ( −1 ) −5 2 = 4 −7 = 4 7

The slope is 4 7 .

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Find the slope of the line that passes through the points ( −2 , 6 ) and ( 1 , 4 ) .

m = 2 3

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Identifying the slope and y- Intercept of a line given an equation

Identify the slope and y- intercept, given the equation y = 3 4 x 4.

As the line is in y = m x + b form, the given line has a slope of m = 3 4 . The y- intercept is b = −4.

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The point-slope formula

Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.

y y 1 = m ( x x 1 )

This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.

The point-slope formula

Given one point and the slope, the point-slope formula will lead to the equation of a line:

y y 1 = m ( x x 1 )

Finding the equation of a line given the slope and one point

Write the equation of the line with slope m = −3 and passing through the point ( 4 , 8 ) . Write the final equation in slope-intercept form.

Using the point-slope formula, substitute −3 for m and the point ( 4 , 8 ) for ( x 1 , y 1 ) .

y y 1 = m ( x x 1 ) y 8 = −3 ( x 4 ) y 8 = −3 x + 12 y = −3 x + 20
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Questions & Answers

find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
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Opoku
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Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
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Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
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AJ
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Atone
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Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
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salma
Practice Key Terms 7

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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